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New Sharp Inequalities in Analysis and Geometry
桂长峰 教授(美国得克萨斯大学圣安东尼奥分校)
2021年6月12日9:30-10:30  闵行数学楼102报告厅

*主持人:叶东 教授

*讲座内容简介:

The classical Moser-Trudinger inequality is a borderline case of Soblolev inequalities and plays an important role in geometric analysis. Aubin in 1979 showed that the best constant in the Moser-Trudinger inequality can be improved by reducing to one half if the functions are restricted to the complement of a three dimensional subspace of the Sobolev space $H^1$, while Onofri in 1982 discovered an elegant optimal form of Moser-Trudinger inequality on sphere. In this talk, I will present new sharp inequalities which are variants of Aubin and Onofri inequalities on the sphere with or without constraints. The main ingredient leading to the above inequalities is a novel geometric inequality: Sphere Covering Inequality, discovered jointly with Amir Moradifam from UC Riverside.

One such inequality, for example, incorporates the mass center deviation (from the origin) into the optimal inequality of Aubin on the sphere which is for functions with mass centeredat the origin. In another view point, this inequality also generalizes to the sphere the Lebedev-Milin inequality and the second inequality in the Szeg\"o limit theorem on the Toeplitz determinants on the circle, which is useful in the study of isospectral compactness for metrics defined on compact surfaces, among other applications.

Efforts have also been made to show similar inequalities in higher dimensions. Among the preliminary results, we have improved Beckner's inequality for axially symmetric functions when the dimension $n=4, 6, 8$. Many questions remain open.



*主讲人简介:

桂长峰教授,美国得克萨斯大学圣安东尼奥分校,任冠名讲座教授。于1984年获北京大学数学士学位,1987年获北京大学硕士学位,1991年在美国明尼苏达大学获博士学位。桂长峰教授曾入选国家级人才计划和海外高层次人才,于2013年当选美国数学会首届会士,获得过IEEE 最佳论文奖、加拿大太平洋数学研究所研究成果奖、加拿大数学中心Andrew Aisensdadt 奖等荣誉。桂长峰教授现致力于非线性偏微分方程的研究,特别是在Allen-Cahn方程的研究、Moser-Trudinger不等式最佳常数的猜想、 De Giorgi 猜想和Gibbons 猜想等方面取得了一系列在国际上有影响的工作,在国际一流数学学术期刊发表论文70余篇,其中包括 Annals of Mathematics, Inventiones Mathematicae, Communications on Pure and Applied Mathematics等顶级期刊。